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Can someone please help and let me know how to approach this question?
My approach is:
Considering statement 1: Prime factors of 12 are: 2,2 & 3. As the GCF of z and 12 is 3, z should be a multiple of 3. Now if its a multiple of 3 it could be 3, 6, 12, 18..... which will be give Yes and No answers to the question and therefore insufficient. But in the book it says it sufficient. Where I am going wrong?
Considering statement 2: Prime factors of 15 are 3 and 5. ...Again I struggle to complete. Can someone please help?
1. The GCF of z and 12 is 3. 2. The GCF of z and 15 is 15.
st-1 is tricky - if the GCF is given as 3, then value of z cannot be 6,12 or any other multiple of 2, because the GCF then would not be 3. So value of Z could be 3, 9, 15, 21 - in all cases the GCF of z and 12 is 3. So Z as per st1 is not divisible by 6.
st 2 - obviously can have values of Z as 15, 30, 45 etc. So Z divisible by 6 may be true or may not be true.
So, if we are given that GCF of z and 12 is 3, what do we know about z. 12=2^2*3
We know that z has at least one "3" in its factor AND z has no factor of 2 because even if there is one factor of 2 present in z, the GCF becomes 2^1*3^1=6, invalidating the statement; you see the point *********************************************** _________________
The LCM of 12 and a contains two 2's. Since the LCM contains each prime factor to the power it appears the MOST, we know that a cannot contain more than two 2's.
LCM of 12 and a contain two 3's. But 12 only contains one 3. The 3^2 factor in the LCM must have come from prime factorization of a. Thus we know that a contains exactly two 3's.
Since a must contain exactly two 3's nd can contain no 2's, one 2 or two 2's a could be
3*3=9 3*3*2=18 3*3*2**2=36.
Thus 9,18 and 36 are three values.
I am struggling to understand the concept.
For LCM you will have to consider all prime factors and maximum powers.
12=2^2*3 a=? 36=2^2*3^3 --------------
What does this tell about a?
LCM always has maximum power of the factor; Thus if LCM is 36 and its factors are 2^2*3^2. It means that a only has a maximum of two distinct prime factors 2 and 3 and the maximum powers of those factors are 2 and 2 respectively.
Now, let's see what 12 tells us; 12=2^2*3 Means; a can have 2^0, 2^1 or 2^2 as its factor because the minimum criteria for 36 to have at least 2^2 has already been taken care by 12. Thus, it really doesn't matter whether a contains 2^2 or not. a may contain 2^0, 2^1 or 2^2. Note a can't contain 2^3 because in 36, maximum power of 2 is 2. Thus, any of the numbers can't have more than 2 2's.
Likewise; let's check for 3. 12 has 1 3. But 36 has two 3's i.e. 3^2 Thus, a must contain 3^2; because 36 is LCM of a and 12. As 12 doesn't have 2 factors of 3. It's become necessary for a to have 2 3's. Thus, a has 3^2. Also, note that a can't contain more than 2 3's because 36 has maximum of 2 3's. Also, a can't contain any other prime factor as 36 has only two distinct factors; 3 and 2.
Now, how many values of a are possible; 2^0*3^2=9 2^1*3^2=18 2^2*3^2=36 **************************************** _________________
Realistic GMAT question would mention that z is a positive integer.
(1) The greatest common factor of z and 12 is 3 --> if z were divisible by 6 (for example 6, 12, 18, ...) then the GCF of z and 12 (which is also divisible by 6) would have been more than 3 (6 or 12) and since the GCF is 3 then z is not divisible by 6. Sufficient.
(2) The greatest common factor of z and 15 is 3 --> if z=3 then the answer is NO but if z=6 then the answer is YES. Two different answers, not sufficient.
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